Problem 1 - A First Course in Probability

Question

A box contains 3 marbles: 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box and then replacing it in the box and drawing a second marble from the box. Describe the sample space. Repeat when the second marble is drawn without replacing the first marble.

Step-by-Step Solution

Answer

The sample space for drawing two marbles with replacement is {RR, RG, RB, GR, GG, GB, BR, BG, BB}, and the sample space for drawing two marbles without replacement is {RG, RB, GR, GB, BR, BG}.

Step 1: Scenario 1: Drawing a marble with replacement

In this scenario, we will draw a marble, replace it, and then draw a second marble. This means that after each drawing, all three marbles are in the box again. The task here is to find all the possible outcomes for this experiment. Step 1: Identify the possible outcomes There are three marbles: red (R), green (G), and blue (B). We will draw two marbles with replacement, which means that for each drawing, there are three possible outcomes. Step 2: Create the sample space We can create the sample space by listing all the possible outcomes for the experiment. Since there are three marbles and we draw two marbles with replacement, there are a total of 3 * 3 = 9 possible outcomes. These are: 1. RR (Red, then Red) 2. RG (Red, then Green) 3. RB (Red, then Blue) 4. GR (Green, then Red) 5. GG (Green, then Green) 6. GB (Green, then Blue) 7. BR (Blue, then Red) 8. BG (Blue, then Green) 9. BB (Blue, then Blue) The sample space for Scenario 1 is {RR, RG, RB, GR, GG, GB, BR, BG, BB}.

Step 2: Scenario 2: Drawing a marble without replacement

In this scenario, we will draw a marble without replacing it, and then draw a second marble. Since we do not replace the first marble, the second draw will be from a box with only two remaining marbles. The task here is to find all the possible outcomes for this experiment. Step 1: Identify the possible outcomes There are three marbles: red (R), green (G), and blue (B). We will draw two marbles without replacement, which means that for the first drawing, there are three possible outcomes, and for the second drawing, there are two possible outcomes. Step 2: Create the sample space We can create the sample space by listing all the possible outcomes for the experiment. Since there are three marbles and we draw two marbles without replacement, there are a total of 3 * 2 = 6 possible outcomes. These are: 1. RG (Red, then Green) 2. RB (Red, then Blue) 3. GR (Green, then Red) 4. GB (Green, then Blue) 5. BR (Blue, then Red) 6. BG (Blue, then Green) The sample space for Scenario 2 is {RG, RB, GR, GB, BR, BG}.